See p. 66 of [1]}; actually in this book, weights are
not used and hence we give an outline of the proof.
First of all, it is easy to check that for any \(\phi =\phi (t)\in W^1_p((0,T))\)  (cf. p.32 of [1]})
\(\sup _{t\le T}|\phi (t)|^p\le N \int ^T_0(|\phi |^p+|\phi ^{\prime }(t)|^p)dt.\) 
